QUANTUM AND CLASSICAL IMPLICATION ALGEBRAS WITH PRIMITIVE IMPLICATIONS

Join in an orthomodular lattice is obtained in the same form for all f ive quantum implications. The form holds for the classical implication in a distributive lattice as well. Even more, the definition added to an ortholattice makes it orthomodular for quantum implications and di stributive for the classical one. Based on this result a quantum impli cation algebra with a single primitive-and in this sense unique-implic ation is formulated. A corresponding classical implication algebra is also formulated. The algebras are shown to be special cases of a unive rsal implication algebra.